The Lagrangian has in fact to depend on reference backgrounds in a quite peculiar way, so that a reference background cannot interact with any other physical field, otherwise its effect would be observable in a laboratory….
Let then Γ’ be any (torsionless) reference connection. Introducing the following relative quantities, which are both tensors:
qμαβ = Γμαβ – Γ’μαβ
wμαβ = uμαβ – u’μαβ —– (1)
For any linear torsionless connection Γ’, the Hilbert-Einstein Lagrangian
LH: J2Lor(m) → ∧om(M)
LH: LH(gαβ, Rαβ)ds = 1/2κ (R – 2∧)√g ds
can be covariantly recast as:
LH = dα(Pβμuαβμ)ds + 1/2κ[gβμ(ΓρβσΓσρμ – ΓαασΓσβμ) – 2∧]√g ds
= dα(Pβμwαβμ)ds + 1/2κ[gβμ(R’βμ + qρβσqσρμ – qαασqσβμ) – 2∧]√g ds —– (2)
The first expression for LH shows that Γ’ (or g’, if Γ’ are assumed a priori to be Christoffel symbols of the reference metric g’) has no dynamics, i.e. field equations for the reference connection are identically satisfied (since any dependence on it is hidden under a divergence). The second expression shows instead that the same Einstein equations for g can be obtained as the Euler-Lagrange equation for the Lagrangian:
L1 = 1/2κ[gβμ(R’βμ + qρβσqσρμ – qαασqσβμ) – 2∧]√g ds —– (3)
which is first order in the dynamical field g and it is covariant since q is a tensor. The two Lagrangians LH and L1, are thence said to be equivalent, since they provide the same field equations.
In order to define the natural theory, we will have to declare our attitude towards the reference field Γ’. One possibility is to mimic the procedure used in Yang-Mills theories, i.e. restrict to variations which keep the reference background fixed. Alternatively we can consider Γ’ (or g’) as a dynamical field exactly as g is, even though the reference is not endowed with a physical meaning. In other words, we consider arbitrary variations and arbitrary transformations even if we declare that g is “observable” and genuinely related to the gravitational field, while Γ’ is not observable and it just sets the reference level of conserved quantities. A further important role played by Γ’ is that it allows covariance of the first order Lagrangian L1, . No first order Lagrangian for Einstein equations exists, in fact, if one does not allow the existence of a reference background field (a connection or something else, e.g. a metric or a tetrad field). To obtain a good and physically sound theory out of the Lagrangian L1, we still have to improve its dependence on the reference background Γ’. For brevity’s sake, let us assume that Γ’ is the Levi-Civita connection of a metric g’ which thence becomes the reference background. Let us also assume (even if this is not at all necessary) that the reference background g’ is Lorentzian. We shall introduce a dynamics for the reference background g’, (thus transforming its Levi-Civita connection into a truly dynamical connection), by considering a new Lagrangian:
L1B = 1/2κ[√g(R – 2∧) – dα(√g gμνwαμν) – √g'(R’ – 2∧)]ds
= 1/2κ[(R’ – 2∧)(√g – √g’) + √g gβμ(qρβσqσρμ – qαασqσβμ)]ds —– (4)
which is obtained from L1 by subtracting the kinetic term (R’ – 2∧) √g’. The field g’ is no longer undetermined by field equations, but it has to be a solution of the variational equations for L1B w. r. t. g, which coincide with Einstein field equations. Why should a reference field, which we pretend not to be observable, obey some field equation? Field equations are here functional to the role that g’ plays in our framework. If g’ has to fix the zero value of conserved quantities of g which are relative to the reference configuration g’ it is thence reasonable to require that g’ is a solution of Einstein equations as well. Under this assumption, in fact, both g and g’ represent a physical situation and relative conserved quantities represent, for example, the energy “spent to go” from the configuration g’ to the configuration g. To be strictly precise, further hypotheses should be made to make the whole matter physically meaningful in concrete situations. In a suitable sense we have to ensure that g’ and g belong to the same equivalence class under some (yet undetermined equivalence relation), e.g. that g’ can be homotopically deformed onto g or that they satisfy some common set of boundary (or asymptotic) conditions.
Considering the Lagrangian L1B as a function of the two dynamical fields g and g’, first order in g and second order in g’. The field g is endowed with a physical meaning ultimately related to the gravitational field, while g’ is not observable and it provides at once covariance and the zero level of conserved quantities. Moreover, deformations will be ordinary (unrestricted) deformations both on g’ and g, and symmetries will drag both g’ and g. Of course, a natural framework has to be absolute to have a sense; any further trick or limitation does eventually destroy the naturality. The Lagrangian L1B is thence a Lagrangian
L1B : J2Lor(M) xM J1Lor(M) → Am(M)